Signals and Systems - Electrical Engineering

486 C H A P T E R 8: Discrete-Time Signals and Systems

For the zero initial condition, it is clear thaty 1 [n]= 2 y[n] when we double the input. One can

also show that superposition holds for this system. For instance, if we let the input be the sum of

the previous inputs,x[n]=u[n]+ 2 u[n]= 3 u[n], and lety 12 [n] be the response when the initial

condition is zero,y 12 [0]=0, we get

y 12 [0]=0.5y 12 [−1]+x[0]+x[−1]= 3

y 12 [1]=0.5y 12 [0]+x[1]+x[0]=7.5

y 12 [2]=0.5y 12 [1]+x[2]+x[1]=9.75

...

showing thaty 12 [n] is the sum of the responses when the inputs areu[n] and 2u[n]. Thus, the

system represented by the given difference equation with a zero initial condition is linear.

Although when the initial condition is−2 or 0, andx[n]=u[n] we cannot find a closed form

for the response, we can see that the response is going toward a final value or a steady-state

response. Assuming that asn→∞we have thatY=y[n]=y[n−1], and sincex[n]=x[n−1]=

1, according to the difference equation the steady-state valueYis found from

Y=0.5Y+2 or Y= 4

independent of the initial condition. Likewise, whenx[n]= 2 u[n], the steady-state solutionYis

obtained fromY=0.5Y+4 orY=8, independent of the initial condition. n

Remarks

n Like in the continuous-time system, to show that a discrete-time system is linear and time invariant an

explicit expression relating the input and the output is needed.

n Although the solution of linear difference equations can be obtained in the time domain, just like with

differential equations, we will see in the next chapter that their solution can also be obtained using the

Z-transform, just like the Laplace transform being used to solve linear differential equations.

8.3.2 Discrete-Time Systems Represented by Difference Equations

As we saw before, a recursive discrete-time system is represented by a difference equation

y[n]=−

N∑− 1

k= 1

aky[n−k]+

M∑− 1

m= 0

bmx[n−m] n≥ 0

initial conditions y[−k],k=1,...,N− 1 (8.31)

If the system is discrete-time, the difference equation naturally characterizes the dynamics of the

system. On the other hand, the difference equation could be the approximation of a differential

equation representing a continuous-time system that is being processed discretely. For instance, to

approximate a second-order differential equation by a difference equation, we could approximate